This regularization has several practical virtues: the regularized problem can be solved efficiently with Sinkhorn's Algorithm [1]}, [2]}, a fast iterative algorithm that only uses matrix-vector arithmetic.
Another advantage, equally important in our setting, is that this approach, as implemented in OTT-JAX [3]} results in fully differentiable quantities.
Namely, assume that the cost matrix \(\mathbf {C}\)  is provided in the form of a differentiable function resulting in entries \(\mathbf {C}_{i,j} \mathrel {\overset{\makebox{[}0pt]{\mbox{\normalfont \tiny \sffamily def}}}{=}}\mathbf {C}(u_i, v_j)\) . Then the gradient of \(\mathrm {L}^{\epsilon }\)  w.r.t. \(u_i\)  exists and is defined everywhere:
\( \forall i, 1 \le i \le n, \left\Vert \frac{\partial \mathrm {L}^{\epsilon }}{\partial u_i}\right\Vert <\infty .\) 
